No you really do not burn calories when you fart although it's good for your health to fart.
There was a claim that you burn around 67 calories if you farted 52 times per day but not everyone farts that much per day.
It is said that you would need to fart at least 52 times per day to burn off 67 calories which would help you lose at least 1 lb of weight.
But since farting actually relaxes your muscles instead of working your muscles you don't actually burn calories or lose weight when you fart.
To lose weight you need to work your muscles through exercise such as walking, running, jogging etc and when you work those muscles then you lose weight.
If farting helped you lose weight then we would all lose weight pretty quickly due to farting and sometimes we fart without even knowing much about it.